Rotational and vibrational dynamics of interstitial …Rotational and vibrational dynamics of interstitial molecular hydrogen T. Yildirim National Institute of Standards and Technology,

PHYSICAL REVIEW B 66, 214301 ~2002!

Rotational and vibrational dynamics of interstitial molecular hydrogen

T. YildirimNational Institute of Standards and Technology, Gaithersburg, Maryland 20899

A. B. HarrisDepartment of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104

and National Institute of Standards and Technology, Gaithersburg, Maryland 20899~Received 13 June 2002; published 6 December 2002!

The calculation of the hindered roton-phonon energy levels of a hydrogen molecule in a confining potentialwith different symmetries is systematized for the case when the rotational angular momentumJ is a goodquantum number. One goal of this program is to interpret the energy-resolved neutron time-of-flight spectrumpreviously obtained for H2C60. This spectrum gives direct information on the energy-level spectrum of H2

molecules confined to the octahedral interstitial sites of solid C60. We treat this problem of coupled transla-tional and orientational degrees of freedom~i! by construction of an effective Hamiltonian to describe thesplitting of the manifold of states characterized by a given value ofJ and having a fixed total number ofphonon excitations,~ii ! by numerical solutions of the coupled translation-rotation problem on a discrete meshof points in position space, and~iii ! by a group theoretical symmetry analysis. Results obtained from thesethree different approaches are mutually consistent. The results of our calculations explain several aspects of theexperimental observations, but show that a truly satisfactory orientational potential for the interaction of an H2

molecule with a surrounding array of C atoms has not yet been developed.

DOI: 10.1103/PhysRevB.66.214301 PACS number~s!: 78.70.Nx, 34.50.Ez, 82.80.Gk, 71.20.Tx

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I. INTRODUCTION

The study of rotational and vibrational dynamics of gumolecules~i.e., CO, O2, H2, etc.! trapped in porous mediasuch as fullerenes, zeolites, and graphite has recently becan active subject both experimentally and theoretically1–5

This is because such studies can yield valuable informaabout the host-guest interactions which could be imporfor several technical applications such as gas separationhydrogen storage.1–3 In particular, hydrogen moleculetrapped in interstitial cavities in solid C60 as well as hydro-gen molecules embedded in nanotube ropes are of intedue to quantum behavior of hydrogen molecules in quzero and one-dimensional sites.2,3,5

In this paper, we develop a detailed analysis of couprotational and vibrational dynamics of a molecular hydrogencapsulated in a solid using numerical, perturbative,group theoretical methods. In particular we will be interesin what one might call the ‘‘weak-coupling limit,’’ when theinteraction between molecular rotations and center-of-mtranslations is weak enough that the rotational angularmentum quantum numberJ is a good quantum number. Thlimit is almost never satisfied except for very light moleculike hydrogen or deuterium. The energy levels of a freetator are

EJ5BJ~J11!, ~1!

whereB5\2/(2I ), I is the moment of inertia of the molecule, andEJ is (2J11)-fold degenerate. For H2 the rota-tional constant 2B has the value 60 cm21, 14.7 meV, orB/k585 K ~and the corresponding values for D2 are half aslarge!, so that the energy separation between differentJ lev-els is large enough that oftenJ is a good quantum numbe

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This is certainly true for solids consisting of these molecuunless the pressure is quite large.~For a review of the prop-erties of the hydrogen molecule and solid hydrogen see R6.!

We have been led to consider this phenomenon in viewan experimental study of energy spectra of H2 and D2 in-serted into the octahedral interstitial sites in solid C60 carriedout by neutron time-of-flight techniques.5 In considering thisphenomenon we should keep in mind the following expemental facts concerning the host solid of C60. The centers ofthe C60 molecules form an fcc lattice.7 At temperature aboveTc , whereTc is about 260 K, the molecules are orientatioally disordered. AtT5Tc long-range orientational orderinoccurs8 and the molecules are ordered into four sublattices

described byPa3 symmetry.9–12 In the orientationally disor-dered phase the local symmetry at the octahedral interssite is indeed that of the point groupOh . In the presence oforientational ordering the symmetry of what was the ‘‘ochedral’’ interstitial site is now reduced to a uniaxial symmtry, specifically that of point groupS6.13 In experiments, hy-drogen molecules are stable in the octahedral interstitialonly for temperatures well belowTc ~where the interstitialsite does not actually have octahedral symmetry!.

While a general understanding of the time-of-flight eperiments was presented,5 some of the finer details of theexperiment remained unexplained. For instance, the shifthe energy associated with ortho-para (J51→J50) conver-sion in the interstitial relative to its value for free moleculwas not understood. Also the feature in the energy gain sptrum at about twice the ortho-para conversion energy wasunambiguously identified. These issues are both addressthis paper. More generally we give a calculation of the crosection for neutron energy loss for comparison with the

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T. YILDIRIM AND A. B. HARRIS PHYSICAL REVIEW B 66, 214301 ~2002!

served time-of-flight spectrum. For that purpose we needonly to consider the cross section for para-ortho conversas compared to phonon creation, but also to calculatephonon excitations of (J51) molecules. These calculationrequire us to develop and implement a scheme for treacoupled translations and rotations. In this paper we presesystematic analysis of the simplest case of this coupwhich occurs when the quantum numberJ characterizingfree rotation remains a good quantum number. In that cthe well-known numerical schemes for solving the transtion problem can be easily extended to include the effecthe coupling to rotations. In addition, we also give analyexpressions obtained by treating this coupling within pertbation theory. As we will see, this analytic development eables us to interpret many of the numerical results in a meingful way. In addition, we analyze in detail variousimplified models which illustrate our group theoreticanalysis of the symmetry present in the system of couptranslations and rotations. This analysis indicates that aments for the degeneracies of coupled translation-rotamodes based on simple classical concepts are incorrecsummary: in this paper we present an analysis based onmerical, perturbative, and group theoretical methods.

Up to now there have not been many theoretical studieenergy levels in such irregular geometries like the octaheinterstitial sites in C60. A notable exception is the work ovan der Avoird and collaborators14 on CO in C60. That workexamined an even more complicated situation in whichrotational and translation degrees of freedom interacstrongly. As a result, the problem was analyzed numericaIn contrast, for the present problem FitzGeraldet al.5 applieda number of analytic and semianalytic techniques to theoretical study the spectra of hydrogen molecules in C60. Thispaper may be regarded as an extension and systematizof their approach.

II. GENERAL FORMULATION

Clearly the first step is to establish a satisfactory potenfor the intercalated hydrogen molecule. This potential fution V(r ,V) gives the energy of a hydrogen molecule whocenter of mass is atr and whose orientation is specified bV[(u,f). A convenient starting point is to use an atomatom potential15 to describe the interaction between eachthe two hydrogen atoms and the atoms in the confining stture. Unless otherwise indicated, all the results reportedthis paper are obtained from the same WS77 potentia15

2A/r 61B exp(2Cr), that is used in Ref. 5~where A55.94 eV Å6, B5678.2 eV, andC53.67 Å21).

In this paper we will mainly consider the octahedral iterstitial site in solid C60, but many of the considerationapply with slight modification to molecules confined withother structures such as single wall carbon nanotubes.16 Thedetermination of the potentialV(r ,V) for H2 in solid C60 isdiscussed in Appendix A. From the numerical evaluationthis potential we have extracted the expansion coefficiewhen it is written in the following canonical form:

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V~r ,V!5V0~r !1 (l 52,4,•••

(m52 l

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Alm~r !Yl

m~V!. ~2!

We assume~and it is generally true! that the orientationalenergies which are relevant are much less than the smaenergy difference between successiveJ levels of a molecule(10B for an ortho molecule and 6B for a para molecule!.Accordingly, we may consider only that part of the potentwhich is diagonal inJ. When the potential is written in theform of Eq. ~2!, it is easy to implement the truncation tterms diagonal inJ. So for a fixed value ofJ we have theHamiltonianHJ as

HJ5p2

2m1V0~r !1BJ~J11!I1 (

l 52,4,•••(

m,m8Al

m2m8~r !

3@ uJm&^JmuYlm2m8~V!uJm8&^Jm8u#

52\2

2m¹21V0~r !1BJ~J11!1VJ~r !, ~3!

whereVJ(r ) is the orientationally dependent part of the ptential ~the terms involvingAl

m) and I is the unit operator.Furthermore, we consider the angular-dependent term inexpansion to be a perturbation on the first term,V0(r ). Foreach value ofJ the HamiltonianHJ will give a manifold ofstates which is the direct product of a manifold correspoing to various numbers of localized phonons being exciwith the manifold of (2J11) states having different valueof mJ . An important simplification is that spherical harmoics with l .2J have no nonzero matrix elements in the mafold of states of angular momentumJ.

Note that apart from the kinetic energy, this Hamiltoniis a strictly local operator. Thus we solve the eigenvaproblem on a discrete mesh of points on a cube centerethe octahedral site when the wave function is requiredvanish on the boundary of the cube. Each edge of the cubtaken to be@2L,L# with mesh point spacing ofdL. In thisscheme the wave function at each mesh point is(2J11)-component vector. We are mainly concerned wthe manifoldJ50 andJ51, in which case the problem inumerically not significantly harder than for a scalar prolem. Even though the resulting matrix size is very large, ita block band matrix and is very sparse. The numerical resreported here were obtained fromL51.65 Å and dL50.075 Å, which requires diagonalization of a matrixn3n where n5273 375). However, we confirmed thatcoarse mesh points withL51.2 Å and dL50.17 Å givesalmost the same results~where the matrix size isn514 739). The large sparse matrix eigenvalue problemsolved using the packageARPACK.17

Since numerical results sometimes do not provide coplete insight into the nature of the solutions, we have aused perturbation theory to understand the results. Inapproach we treatVJ(r ) in Eq. ~3! as the perturbation. Theunperturbed problem, apart from the additive energyBJ(J11) is thus that for translations of the spherical (J50) mol-ecule. This spectrum is not too different from that of a thredimensional harmonic oscillator. Accordingly, to qualit

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ROTATIONAL AND VIBRATIONAL DYNAMICS O F . . . PHYSICAL REVIEW B 66, 214301 ~2002!

tively interpret our more accurate numerical results,apply perturbation theory in which we develop an effectHamiltonian18 to describe the splitting of this manifolwhich is characterized by a value ofJ and of N, the totalnumber of phonon excitations.~The perturbative effects duto coupling between manifolds of differentJ is negligiblysmall for hydrogen in C60.5! This effective Hamiltonian is amatrix of dimensionalityD, where D5(2J11)(N11)(N12)/2 and schematically is of the form

H~N,J!5BJ~J11!I1Hphonon1VJDIAG2VJ

OFF1

E VJOFF,

~4!

whereHphonon gives the energy of the various states withtotal of N phonons. These energies are just those calculfor a (J50) molecule. AlsoVJ

DIAG is the part ofVJ which isdiagonalwith respect to the number of phonons, VJ

OFF is thepart of VJ which is off-diagonalwith respect to the numbeof phonons, andE is the change in phonon energy causedVJ

OFF.This effective Hamiltonian is defined by its matrix el

ments as

^N,a;J,M uH~J,N!uN,a8;J,M 8&

5@BJ~J11!1EN,a#da,a8dM ,M8

1(l 51

J

^N,auA2lM2M8~r !uN,a8&

3^JMuY2lM2M8~V!uJM8&1 (

N8ÞN(

l ,l 851

J

(m

(b51

kN8

3@EN,a2EN8,b#21^N,auA2lM2m~r !uN8,b&

3^N8,buA2l 8m2M8~r !uN,a8&^JMuY2l

M2m~V!uJm&

3^JmuY2l 8m2M8~V!uJM8&, ~5!

wherekN5(N11)(N12)/2 and the states withN phononsare labeledN,a, wherea runs from 1 tokN .

We now briefly discuss how theAlm’s of Eq. ~2! are ob-

tained from the atom-atom potential between each H aand each carbon atom. Here we will assume that the atatom potential is of the formF(ur i2rHu), wherer i and rHare the displacements of thei th carbon and of the H atomrespectively, relative to the center of the H2 molecule. Forthis form of potential, we show in Appendix A that

A2m52p(

iY2

m~ r i !* E21

1

~3x221!

3F~@r i22 1

4 r21rxri #1/2!dx, ~6!

wherer is the separation between H atoms in the H2 mol-ecule and the sum overi is over all relevant neighboringcarbon atoms. It is instructive to expand this expressionr/r i , which yields

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m~ r i !1o~r4/r i4!, ~7!

where F9 and F8 are the second and first derivativesF(r i).

This expansion is good enough to reproduce most ofresults discussed in this paper. We note that theAl

m’s ~i.e., theorientational potential! are zero for a harmonic [email protected].,F(r i)5 1

2 kri2] because the prefactor (F92F8/r i) is zero.

This can be also seen easily as follows. Assuming an atatom potential between each H atom and the C atoms inadjacent C60 molecules, we may write the potential of an H2molecule as

V~r ;V!5Va~r1 12 rn!1Va~r2 1

2 rn!, ~8!

whereVa is the potential of a single atom due to the entoctahedral cage in which it is confined,n is a unit vectoralong the axis of the molecule, andr is the separation between atoms in the molecule. As we shall see, the totaltential is nearly isotropic. So we write

Va~r !5 12 kr21dr 4. ~9!

When we substitute this into Eq.~8!, we obtain the result

V~r ;V!5V0~r !12d~r 2r2cos2u r ,n2 12 !, ~10!

whereu r ,n is the angle between the vectorsr andn andV0 isindependent ofu r ,n . The point is that the orientationally dependent part of the interaction depends on the anharmoity: for a purely harmonic and isotropic interactionVa , thetotal potential energy is independent of the molecular oritation. Thus we expect rotation-translation coupling toweak. On the other hand in nanotubes, where the quadterm isanisotropic, this coupling will be more important.16

III. ENERGY SPECTRUM OF A „JÄ0… H2 MOLECULE

We start by considering the eigenvalue spectrum ofH0 inwhich the orientational dependence of the potential isglected. In this approximation, apart from the additive costant BJ(J11), the total energy~rotational plus transla-tional! is the same as that of a (J50) molecule. For mostpurposes a (J50) molecule may be considered to bespherical molecule because the orientational wave funcY0

0(V) is uniform over all orientations. Each eigenfunctioof H0(J) is the product of a rotational function taken frothe manifold of 2J11 degenerate orientational wave funtions and a translational wave function which representseigenfunction for a spherical molecule confined to a caThese translational wave functions satisfy

H0ck~r !5S p2

2m1V0~r ! Dck~r !5Ekck~r !, ~11!

whereV0(r ) is the potential discussed in Appendix A. Thindexk labels states which we might otherwise label by thrindices, each quantum number characterizing the numbeexcitations in each direction. Note that these unperturbed

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lutions do not involve the coupling between rotations antranslations. As discussed above, these eigenfunctionsobtained by converting the continuum equation~11! into adiscrete equation on a mesh of points and solving the resing matrix eigenvalue problem using a sparse maroutine.17

Since it happens that the energy levels and eigenfunctwe obtained numerically are not qualitatively different frothose of a spherical harmonic oscillator, we first studyenergy spectrum as perturbationsd, k, and l are sequen-tially turned on in the following potential:

V~r !5 12 kr21dr 41k~x41y41z42 3

5 r 4!

1l~xy1yz1zx!. ~12!

Figure 1 shows the evolution of the energy spectrum asturbations are sequentially introduced which take the sphcal harmonic oscillator into the actual lower symmetry o

FIG. 1. Energy levels of a spherical (J50) H2 molecule con-fined in various ways. Here~g! is the degeneracy and the symmetlabels are given. Left: The molecule is in a spherical harmopotentialV(r )5

12 k(x21y21z2) or an anharmonic spherically sym

metric potential~i.e., a generic spherically symmetric potential!. Fora harmonic spherically symmetric potential the energy dependson N, the total number of phonon excitations in the oscillators alothe three coordinate directions. For a spherically symmetric potial eigenstates are characterized by their total orbital angularmentumL. Center: the molecule is in a potential appropriate tooctahedral interstitial site of orientationally disordered (Fm3m)solid C60. Right: the molecule is in a potential appropriate to t

octahedral interstitial site of orientationally ordered (Pa3) solidC60, in which case the site symmetry isS6. The potentials used fothe interstitial cases are discussed in Appendix A.

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molecule in an octahedral interstitial site. In the left-mopanel we show the energy levels for a spherical harmooscillator, with \v adjusted to correspond to the singlphonon levels of H2 in an octahedral interstitial site in C60.Note that the levels are highly degenerate because the endepends only on the total number of quanta. The symmof the Hamiltonian is U(3), the group of unitary three-dimensional matrices. We now add to this potential an anhmonic term of the formdr 4. This perturbation lowers thesymmetry to that of the rotation group in three dimensioAs is well known, each eigenfunction in a generic sphecally symmetric potential can be labeled by the magnitudethe orbital angular momentumL. Thus the single phononlevels are unsplit by this anharmonic perturbation andnow labeled as angular momentumL51 states, whereas thtwo phonon levels split into a manifold of fiveL52 statesand oneL50 state and similarly for states with more thatwo phonons. In Fig. 1 we have taken the constantd to bethat which best describes the anharmonicity of H2 in C60.The energies of the perturbed levels are given in Table I

Next, we consider what happens when the spherical oslator potential is augmented by a cubic symmetry potentiathe form k(x41y41z42 3

5 r 4). This potential is appropriatefor a spherical molecule in an octahedral interstitial whenC60 are orientationally disordered and have anFm3m crystalstructure.7 The degeneracy associated with spherical symmtry is lifted,19 but as shown here one retains cubic symmeso the three one-phonon states which transform asx, y, andzare degenerate. The two-phonon states are of three diffesymmetries. One (t2g) transforms likexy, xz, andyz. This isthe lowest level. The next highest level is thes-wave sym-metric combination which transforms likex21y21z2. Thenone has a doublet ofd-wave (e2g) symmetry. This classifi-cation scheme is continued in the higher-energy levAlthough we are not dealing with harmonic phonons,is still useful to consider manifolds characterized by tquantum numbersJ and N, which are respectively the

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TABLE I. Effect of a spherical symmetric perturbation onU3

states. TheU3 states are characterized byN, the total number ofharmonic phonons. Wave functions in a spherical potential are cacterized by angular momentumK. Here we give the effect of theperturbationD(r /s)4, where^r 2&53s2 for the isotropic harmonicoscillator in three spatial dimensions. In the last column we giveshift in the average energy of the multiplet of states of a given vaof N.

N N,K Energy Avg.E

0 ~0,0! 15 D 15 D

1 ~1,1! 35 D 35 D

2 ~2,0! 75 D 65 D

~2,2! 63 D

3 ~3,1! 119 D 105 D

~3,3! 99 D

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TABLE II. Energy-level systematics for a (J50) molecule in an octahedral interstitial site of C60. Herewe show the removal of degeneracy from a manifold of initial symmetryI to manifolds of final symmetryFdue to a perturbationV, as calculated in lowest-order perturbation theory. HereN is the total number ofphonons,K is the angular momentum, and the other group theoretical labels are as in Fig. 1. We give aeigenfunctionc to illustrate the symmetry. Hered is the degeneracy~Deg.! of the manifold ands25^x2&5^y2&5^z2&. Here the coordinate axes coincide with the cubic@100# directions.

I c V c F Deg. Energy

N52,K52 r 2Y2M(V) k(x41y41z42

35 r 4) (x22y2) Eg 2 36

5 ks4

xy T2g 3 2245 ks4

N53,K53 r 3Y3M(V) k(x41y41z42

35 r 4) (x32

35 xr2) T1u 3 36

5 ks4

x(y22z2) T2u 3 2125 ks4

xyz A2u 1 2725 ks4

N51,K51,T1u rY1M(V) l(xy1yz1zx) Eu 2 2ls2

Au 1 2ls2

N52,K52,T2g xy l(xy1yz1zx) Eg 2 2ls2

Ag 1 2ls2

N53,K53,T2u x(y22z2) l(xy1yz1zx) Eu 2 32 ls2

Au 1 23ls2

N53,K53,T1u (x3235 xr2) l(xy1yz1zx) Eu 2 3

10ls2

Au 1 235 ls2

N53,K51,T1u x(r 2213 s2) l(xy1yz1zx) Eu 2 3ls2

Au 1 26ls2

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rotational angular momentum and the total numberphonons, at least up toN53. Quantitative results are givein Table II.

Finally, in the right-most panel of Fig. 1 we show thfurther reduction in degeneracy which occurs when the ohedral interstitial is surrounded by C60 molecules which havethe long-range order associated with thePa3 crystalstructure.9–12 In this case, each interstitial is uniaxial~withsymmetryS6) rather than octahedral. Accordingly, we intrduce a potential of the forml(xy1yz1zx)[ 1

2 l(3j2

2r 2), where thej axis is taken to lie along the threefolaxis of the interstitial site. There are four symmetry relainterstitial sites, each of which has its threefold axis alondifferent @1,1,1# direction. The resolution of degeneracythe presence of this uniaxial perturbation is also givenTable II. In all these cases, no interactions between rotatand translations are involved.

We have solved the eigenvalue problem of Eq.~11! on amesh of points and obtained the results given in TableResults labeled ‘‘Octahedral’’ are those for the orientatioally disordered phase, where each C60 molecule is replacedby a sphere of carbon atoms as is discussed in Ref. 5. Sthese numerical results lead to manifolds of energy levassociated with a given number of phonons and the deeracies of these manifolds are as expected from our gendiscussion above, we conclude that the potential seenspherical H2 molecule in the low-lying phonon levels is novery different from that of a spherical harmonic oscillatoHowever, as noted in Ref. 5, the effective harmonic potenmust be taken to be a self-consistently renormalized poteto take account of the larger zero-point motion.

21430

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IV. ENERGY SPECTRUM OF A JÄ1 MOLECULE

We now discuss the energy spectrum of an ortho molecwith (J51). As we have seen for (J50) molecules, our

TABLE III. Phonon levels for a (J50) H2 molecule in an oc-tahedral site for orientationally ordered and disordered C20, respec-tively. Energies~meV! are with respect to the ground-state enerE0,1. The symmetry of each manifold of degenerate levels canread from Fig. 1.

EN,a Octahedral Pa3 (S6)

N a

1 1 14.38 13.161 2,3 14.38 14.47

2 1 28.26 26.692 2,3 28.26 27.492 4 30.69 30.412 5,6 31.73 31.39

3 1 41.62 40.103 2,3 44.39 42.40a

3 4 44.39 42.42a

3 5,6 45.23 44.403 7 45.23 45.833 8,9 50.07 49.363 10 50.07 49.85

aThese energies are accidentally almost identical. However, grtheory indicates that these levels are generically nondegenera

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numerical results indicate that forN up to, say, 3, one canclearly identify the manifold ofN phonons. We thereforediscuss the systematics of these manifolds.

A. Zero-phonon manifold

We first consider the case ofJ51 with N50 phonons.This manifold is described by the effective Hamiltonian

H5~2B1DE!I1d~Jz22 2

3 !. ~13!

The splittingd must be zero when C60 is orientationally dis-ordered. From Eq.~5! one sees that because the spherharmonics are traceless, the average energy shiftDE hasnonzero contributions only from terms which involve copling to excited phonon states.~In Ref. 5 a negligibly smallshift was found due to off-diagonal effects inJ which weignore here.! From Eq.~5! we find that

DE521

3 (NÞ0,a

EN,a21 (

m,tu^0,1uA2

t~r !uN,a&u2u

3^1~m1t!uY2t~V!u1m&u2. ~14!

To implement this equation, we first constructAlm(r ) as dis-

cussed in Eq.~6!. Then matrix elements ofA2M(r ) are taken

between phonon states for aJ50 molecule which we ob-tained previously and which are labeledN,a (u0,1& being thephonon ground state!. Thereby we obtained the results givein Tables IV and V. In Eq.~14! the matrix elements ospherical harmonicsY2

M(V) are taken between orientationstates labeled byJ andJz . To evaluateDE we use

TABLE IV. Matrix elements of NauA2t(r )u01& ~in meV! for H2

in octahedral andS6 potential, respectively. Elements not listed aexpected to be zero by symmetry. Numerically such elements wfound to be very small. For this table the wave functions witheach degenerate manifold were chosen to make the matrix elem

of A2t(r ) as simple as possible. ForPa3 symmetry, thez axis is

taken to be the local threefold axis. Therefore the octahedral w

functions are not necessarily identical to thePa3 wave functions.

^NauA2t(r )u01& Octahedrala Pa3 (S6)

N a t

0 1 0 ~0,0! (21.286,0)2 1 0 ~0,0! (20.506,0)2 1 1 (a,0) ~0,0!2 1 2 ~0,0! ~0,0!2 2 1 (0,a) (20.176,20.106)2 3 2 (0,a) (2.413,22.388)2 4 0 ~0,0! (20.224,0)2 5 0 ~0,0! ~0,0!2 5 2 (b,0) (26.500,21.754)2 6 0 (A2b,0) ~0,0!2 6 1 ~0,0! (28.862,21.400)

aOur numerical results givea50.859 andb55.662~in meV!.

21430

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u^1~m1t!uY2t~V!u1m&u25

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so that

DE521

10p (NÞ0,a

EN,a21 (

tu^0,1uA2

t~r !uN,a&u2. ~16!

In Appendix B we give a model calculation of an H2molecule in a spherical cavity from which we evaluate E~16! to give DE520.14 meV. In this calculation the translational wave functions are assumed to be those of amonic oscillator with^r 2&50.1875 Å2. As noted, the resultis very sensitive to the value used for^r 2&. For octahedralsymmetry~i.e., for orientationally disordered C60) we evalu-ate Eq.~16! using the data in Table IV. Thereby we findshift DE520.133 meV. The same approach using our nmerical solutions for the phonon states of a (J50) moleculefor the orientationally orderedPa3 phase yields the resultDE520.141 meV, compared to the experimental valu5

DE520.35 meV. Again we mention that a small changeparameters could easily lead to a much larger calculavalue ofDE. From the numerical solution for the three component wave function of a (J51) molecule on a mesh opoints, we obtained the valueDE520.16 for the Pa3phase. The various numerical results forDE are summarizedin Table VI.

From Eq.~5! we also find the splitting~in thePa3 phase!to be

re

nts

ve

TABLE V. Nonzero matrix elements of1auA2t(r )u1b& ~in

meV! for H2 in octahedral and S6 potential, respectively. We alsonote that ^1auA2

t(r )u1b&5^1buA2t(r )u1a& and ^1auA2

2t(r )u1b&5(21)t^1auA2

t(r )u1b&* . For the octahedral symmetry, the wavfunctions within each degenerate manifold were chosen to makematrix elements ofA2

t(r ) as simple as possible.

^1auA2t(r )u1b& Octahedral Pa3 (S6)

t a b

0 1 1 (23.853,0) (22.020,0)0 2 2 (23.853,0) (21.326,0)0 3 3 ~7.708,0! (21.326,0)1 1 2 ~0,0! (22.109,21.131)1 1 3 ~0,0.881! (1.137,22.104)1 2 2 ~0,0! ~1.534,2.440!1 2 3 (20.881,0) (2.435,21.533)1 3 3 ~0,0! (21.531,22.442)

2 1 1 (24.119,0) ~0,0!2 1 2 ~0,0.880! (20.225,23.022)2 1 3 ~0,0! (23.022,0.226)2 2 2 ~0,0! ~0.560,0.835!2 2 3 ~0,0! (20.830,0.556)2 3 3 ~0,0! (20.565,20.831)

1-6


TABLE VI. Shift of the center of gravity~CG! and splitting~in meV! of the (J51,N50) manifold whenthe nominally octahedral site has octahedral andS6 symmetry.

Our calculations Experimentd

Quantity Octahedral (Oh) Pa3 (S6) Pa3 (S6)

Shift of CGa 20.134 20.141 0.35Shift of CGb 20.16 0.35Splitting first orderc 0 0.487Second orderc 0 20.010Totalc 0 0.477 0.70Totalb 0 0.46 0.70

aPerturbation result of Eq.~16!.bObtained by direct diagonalization of Eq.~3!.cPerturbation result of Eq.~17!.dFrom Ref. 5.

foixtoaln

acynothiseckforsh

tion

d523^0,1uA2

0u0,1&

A20p1

3

20p (e

E~e!21@ u^0,1uA20~r !ue&u2

1u^0,1uA21~r !ue&u222u^0,1uA2

2~r !ue&u2#, ~17!

where the quantization axis is taken to lie along the threeaxis of symmetry of the interstitial site. Using the matrelements given in Table IV, we find that the contributionthe splittingd comes almost exclusively from the diagonterm ^0,1uA2

0u0,1& and we obtain the results listed iTable VI.

B. One-phonon manifold

1. Numerical results

Next we consider the manifoldJ51 with N51 phonon.Again only Yl

m with l 52 contributes, so that we may write

H~1,1!am;a8m8[âmuH~N51,J51!ua8m8&

5@2B1E01\va#dm,m8da,a8

1^1auA2m2m8~r !u1a8&

21430

ld

3^1muY2m2m8~V!u1m8&

2 (N8Þ1

(a951

kN8

(m9

1

EN8,a92E1,a

3^1auA2m2m9~r !uN8a9&

3^N8a9uA2m92m8~r !u1a8&

3^1muY2m2m9~V!u1m9&

3^1m9uY2m92m8~V!u1m8&. ~18!

This is a 939 matrix, which gives the 9 (J51,N51) levels.Since the first term gives rise to the removal of degenerexpected from group theoretical considerations, we didinclude the second term in our numerical evaluations. Tprocedure was sufficiently accurate to provide a useful chon the validity of the more accurate numerical solutionsthe three-component wave functions of our set of mepoints. In Table VII these numerical results~‘‘full mesh’’ ! aregiven and are compared to the results using perturba

d

TABLE VII. Energy ~in meV! of J51 single-phonon states where zero-of-energy is taken to be 2B.

Octahedral (Oh) Pa3 (S6)

C Full mesha Perturbationb C Full mesha Perturbationb

T1g 13.11 13.14 Ag 12.59 12.52T1g 13.11 13.14 Eg 12.82 12.87T1g 13.11 13.14 Eg 12.82 12.87T2g 13.6 13.68 Eg 13.47 13.51T2g 13.6 13.68 Eg 13.47 13.51T2g 13.6 13.68 Ag 14.20 14.16Ag 15.61 15.78 Ag 15.75 15.31Eg 16.46. 16.60 Eg 16.60 15.79Eg 16.46. 16.60 Eg 16.60 15.79

aSolution to Eq.~3! for the three component wave function on a mesh of points.bSolution to Eq.~5! using wave functions and energies for aJ50 molecule as previously determinenumerically on a mesh of points.

1-7

e

inintio

u-e

atoo

pat

ri

in

ngn

ionn

o

ouyreulitn-isia

esty-

blta

rep-for

hat

ndas-n inis

lin-nc-lta-.p-hishat

arey isthemedthe

rall

w-


theory, as in Eq.~18!. As can be seen, the two approachyield quite compatible results.

2. Qualitative remarks

Some additional comments on Eq.~18! are in order. Thefirst line of this equation gives the energy at first orderperturbation theory. At this order the wave function remaa product of the spatial ground-state spatial wave funcfor a J50 molecule times aJ51 rotational wave function.At this level of approximation there is no dynamical copling between translation and rotation. In second-order pturbation theory we see that admixtures of two phonon stwhich are multiplied by different rotational states are intrduced. For example, consider the situation when the mecule is in a uniaxial symmetry site and letuX&, uY&, anduZ&be theJ51 states for which respectivelyJx , Jy , andJz arezero. If, for simplicity, we assume that the unperturbed stial wave function is spherically symmetric, then the stawhich without perturbation was

C0uZ&e2r 2/(4s2), ~19!

whereC0 is a normalization constant, is now

C1uZ&e2r 2/4s21C2uX&zxe2r 2/(4s2)1C2uY&zye2r 2/(4s2),

~20!

whereC1'C0 andC2 is small compared toC1. The point isthat this formulation allows the molecule to change its oentational state as it translates. For H2 in C60 this effect issmall, however, in less symmetrical cavities, asnanotubes,16 this effect can become more important.

3. Group theoretical analysis

In Fig. 2 we show the influence of roton-phonon coupliand local site symmetry on the energy spectrum of the ophonon (J51) manifold. At the far left we start from thecase of highest symmetry when the phonon and rotatseparately have complete rotational invariance andphonon-roton coupling is present. In this case the manifof nine states@three one-phonon states three (J51)states# is completely degenerate. When roton-phonon cpling is included ~but the environment is still sphericallsymmetric! we have overall rotational invariance and thesulting eigenstates are characterized by their total angmomentumK. The roton-phonon coupling causes states wdifferent K to have different energy, as illustrated in Appedix B 3. ~The size of the splitting shown in the figureadjusted to agree with the center of gravity of the approprlevels for cubic site symmetry.!

The two right-hand columns pertain to the situation wha (J51) hydrogen molecule occupies the octahedral intertial site of C60. When the C60 molecules are orientationalldisordered the interstitial site hasOh symmetry and we consider that case first. Use of the character tables for theOhgroup indicates that the original nine dimensional reducirepresentationG is decomposed into irreducible represention of theOh group as

21430

s

sn

r-es-l-

-e

-

e-

so

ld

-

-arh

te

ni-

e-

G5T2g% T1g% Eg% Ag , ~21!

and the basis functions associated with these irreducibleresentations are given in Table VIII. As mentioned above,temperatures below about 260 K, the C60 molecules orderinto a structure of crystal symmetryPa3,9–12 in which casethe formerly octahedral interstitial has the lowerS6symmetry.13 Use of the relevant character table shows tnow

G53Ag% 3Eg% 3Eg* , ~22!

whereEg is a complex one-dimensional representation aEg* is its complex conjugate partner. The basis functionssociated with these irreducible representations are giveTable VIII. The most important conclusion from this analysis that the energy eigenfunctions arenot simply products oftranslational and rotation wave functions, but instead areear combinations of such products. This type of wave fution reflects the fact that symmetry operations act simuneously on the position and the orientation of a molecule

To emphasize this fact we give, in Fig. 3, a pictorial reresentations of the translation-rotation wave functions. Trepresentation is to be interpreted as follows. We know tthe rotational wave functions for a freeJ51 molecule can be

FIG. 2. Removal of degeneracy as roton-phonon interactionsintroduced and the site symmetry is lowered. The degeneracindicated by the number in parentheses. At the far left is showncompletely degenerate level when spherical symmetry is assuand no roton-phonon coupling is present. The next panel showseffect of allowing roton-phonon interactions but preserving ovespherical symmetry. HereK is the total~orbital plus orientational!angular momentum. In the next panel spherical symmetry is loered to octahedral symmetry which is appropriate for H2 in theoctahedral interstitial site in orientationally disordered C60. The farright panel ~and the energy scale! applies to the case of H2 inorientationally ordered C60 in which case the site symmetry isS6.

1-8


TABLE VIII. Basis functions within the one-phonon (J51) manifold. Herex, y, and z are the one-phonon states with a single excitation in the phonon associated with thex, y, andz direction, respectively.a

In terms of the mJ states ~denoted umJ&) within (J51) we have X[(u21&2u1&)/A2, Y[ i (u1&1u21&)/A2, andZ[u0&.

Oh symmetrya

T2g (xZ1zX), (yZ1zY), (xY1yX)T1g (xZ2zX), (yZ2zY), (xY2yX)Eg (2zZ2yY2xX), (xX2yY)Ag (xX1yY1zZ)

Pa3 symmetryb

Eg andEg* (xZ1zX,yZ1zY), (xZ2zX,yZ2zY), (xX2yY,xY1yX)

Ag zZ, xX1yY, xY2yX

aThe x, y, andz directions are taken to coincide with the fourfold axis ofOh .bThe z direction coincides with the threefold axis ofS6.

thtaa

orchhn

o-fF

laa

onergon

thtion

haeo

the

ts anithl

toed

ood

taken to be analogs ofpx , py , andpz functions and we willlabel theserotational wave functions asX, Y, or Z. For in-stance uX&;sinu cosf, uY&;sinu sinf, and uZ&;cosu.These wave functions have two lobes, one positive, the onegative, aligned along the axis associated with their slabel. When such a rotational function is multiplied byone-phonon function in thea direction, (ux& denotes a wavefunction for a single-phonon excitation in thex direction!,the total wave function will be an odd function ofa. Thusthe wave functionuxX& is an odd function ofx and is there-fore depicted by twopx functions, one at positivex andanother at2x, with the signs of the two lobes changed. Fsimplicity in the figures we show only those functions whihave appropriate dependence in the plane of the paper, wis taken to be thex-y plane. At the upper left we show axy-like function. It has two othert2g symmetry partnerswhich arexz-like andyz-like. At the upper right we show oneof the twoEg functions which isx22y2-like. These fivet2gandEg functions comprise the manifold of total angular mmentumK52 states. Within spherical symmetry all five othese states are degenerate in energy. In the lower left of3 we show thez-like function of t1g symmetry. Its other twopartners are obtained by cyclically permutingx, y, and z.These three functions comprise the manifold of total angumomentumK51 states, which transform under rotation asvector. Finally at the lower right we show the angular mmentumK50 state. Thus in spherical symmetry, the niJ51 single phonon states give rise to three distinct enemanifolds which have degeneracies 1, 3, and 5, corresping respectively to total angular momentumK50, K51,andK52.

The simplest classical arguments do not reproduceabove results. For instance, one might argue that translacan occur equivalently along either of three equivalent codinate axes. In each case, one can have the molecule oriealong the axis of translational motion or perpendicular to taxis. This argument would suggest that the nine levels brinto a threefold degenerate energy level in which the m

21430

erte

ich

ig.

r

-

yd-

eonr-tedt

akl-

FIG. 3. Translation-rotation wave functions for a (J51) H2

molecule in an octahedral interstitial site with one phonon. Hereplane of the paper is thex-y plane and for simplicity only thedependence in this plane is depicted. Each figure eight represenuX& or uY& orientational wave function and the sign associated weach lobe of thisp-like function is indicated. Each orientationawave function is multiplied by a translational wave functionux&,uy&, or uz&, where for instanceux&;x exp@2 1

4(x/s)2#. The presenceof a phonon in ther a coordinate thus causes the wave functionbe an odd function ofr a , as one sees in the diagrams. As indicatin Fig. 2, the total angular momentumK, which is the sum of theangular momentum of the phonon and that of rotation, is a gquantum number whose value is indicated. Top, left: aK52, T2g

function; top right: aK52, Eg function; bottom left: aK51, T1g

function; and bottom right aK50, Ag function.

1-9

howem

wa

obc

a.onleH

inb

nw

wom

u

in

ne

d toing,

o

um

s

sses.

ra-

tron-od

nal

on,

lto

ssns

tion1nsof

thtron

verve

bein(

thetio

c-


ecule is oriented longitudinally and a sixfold level in whicthe molecule is oriented transversely. This discussion shthat it is essential to treat the translation-rotation problquantum mechanically to get the correct degeneracy.

V. NEUTRON SPECTRUM

In the experimental study of Fitzgeraldet al.,5 the neutronenergy-loss spectrum of H2 trapped in C60 was measuredwith an energy resolution of 0.3 meV. The spectrum shosurprisingly rich features. However, the origin of these fetures was not successfully identified in detail. Since theserved neutron spectrum is a direct probe of the intermolelar potential between H2 molecules and C60 host lattice, it isvery important to see if available atom-atom potentials cgive a spectrum which is similar to the experimental datasuitable analysis of the high-resolution inelastic neutrscattering data in Ref. 5 should, in principle, give a detaiinformation about the intermolecular potential between2molecules and the host lattice.

Figure 4 illustrates several possible transitions, involvboth rotational and vibrational excitations, that could be oserved in a neutron-scattering experiment for H2 in solidC60. In order to estimate the intensities of these transitioand the corresponding neutron spectrum, in Appendix Cderive the inelastic neutron cross section for trapped H2 mol-ecules in a powder sample at low temperature. Belowdiscuss the contribution to the total neutron spectrum freach of these transitions, labeled asTA , . . . ,TE , and thencompare the calculated spectrum with experimental dataing various atom-atom potentials.

We start with the transitions involving phonon creationpara hydrogen, as shown byTD in Fig. 4. Because of thespin-dependent interaction between the proton and the

FIG. 4. A schematic representation of possible transitionstween the rotation-phonon energy levels that could be observedneutron-scattering experiment. At low temperature, only theJ50,N50) and (J51,N50) states are populated and thereforetransitionTC can not be observed at low temperature. The transiTD is proportional to the coherent cross section of H2 and thereforevery small. The transitionsTB andTA have comparable cross setions ~see text for details!.

21430

s

s--

u-

nA-d

g-

se

e

s-

u-

tron, processes in which a para molecule is not convertean ortho molecule are forbidden, or more correctly speakare proportional to the coherent cross sectionb, which isvery small compared to incoherent cross sectionb8. Hencethe transitionTD will not have a noticeable contribution tthe total cross section.

We next discuss the contribution to the total spectrfrom processes in which either a (J50) molecule is con-verted to (J51) molecule~para-ortho conversion, labeled aTA in Fig. 4! or a single phonon is created~as shown byTB).Our calculations presented so far indicate that both procewill give features around 14 meV in the neutron spectrum

In Appendix C we find that the cross section due to paortho conversion~indicated by the subscript 0→1) is givenby

]2s

]V]E D0→1

53

4N

k8

k~12x!Fb8 j 1S 1

2kr D G2

e22W(k)

3(m

d@EL2~Ec1Em!#, ~23!

whereN is the total number of H2 molecules,k (k8) is wavevector of the incident~scattered! neutron,k5k82k, x is thefraction of H2 molecules which are ortho~oddJ) molecules,r is the separation between protons in the H2 molecule,b8 isthe spin-dependent cross section in the proton-neupseudopotential,j n is the nth-order spherical Bessel function, andW(k) is the Debye-Waller factor which we take tbe 1

3 k2û2&. Also, EL is the energy loss of the neutron, anEc1Em is the para-ortho conversion energy when the fistate of the ortho hasJz5m.

Similarly, the cross section due to ortho-para conversi]2s/]V]E)1→0 has the same expression as Eq.~23! but nowthe factor (12x) is replaced byx. Hence the ratio of the totacross section for ortho to para conversion to that of paraortho conversion is (12x) to x, wherex is the ortho concen-tration. Normally the ratio of energy gain to energy loss crosections follows the Boltzmann factor. Here, the populatioare set byx rather than by the temperature.

We now discuss the cross section due to phonon creaon a (J51) molecule ~indicated here by the subscript→1), which is calculated in Appendix C. These transitioare shown asTB in Fig. 4. The result requires a knowledgethe translation-rotation wave function of the H2 molecule.We find that

]2s

]V]E D1→1

5Nxk8

kb82(

n51

4

S1→1,n(1) , ~24!

where the cross sectionS1→1,n(1) are given in Eqs.~C34!,

~C35!, and~C43! of Appendix C.In Fig. 5 TB represents the transitions from the (J51,N

50) levels to the manifold of nine energy levels of (J51,N51). Accordingly, we expect several transitions winonzero amplitude and thus rich features in the total neucross section.

Also one may consider the cross section integrated oenergy, which is a useful quantity to indicate the relati

-a

n

1-10

rfo

umth

th

he

rgyross

a

ore

theso-

en-re-ownatn-

datacal-fore

lit-e isdy

ppo-lso

(

iet

uri

s

5

77ro-is

as


strength of the different transitions discussed above. Thetio r P of the integrated neutron energy-loss cross sectionphonon creation~processTB in Fig. 4! divided by that forpara-ortho conversion~processTA in Fig. 4! was found inAppendix C to be

r P52

27k2û2&

x@ j 0~ 12 kr!212 j 2~ 1

2 kr!2#

~12x! j 1~ 12 kr!2

. ~25!

This ratio is plotted as a function ofk for x53/4 in Fig. 6.Since this ratio is of order unity, the energy-loss spectrwill display features due to both phonons and para-orconversion.

In Appendix C we also calculate the zero-phonon orcross section for the transition shown asTE in Fig. 4. Sincethe (J51) levels are in thermal equilibrium, in this case t

FIG. 5. The calculated transition probabilities from theJ51,N50) levels to the ninefold manifold of (J51,N51) levels atT54 K. The energies of the levels are given in Table VII~underfull mesh!. For each pair of energies these transition probabilitrepresent the appropriate sum over degenerate levels. Notethere are at least eight transitions with comparable probability, sgesting that rich features could be observed in a neutron-scatteexperiment.

FIG. 6. The solid curve is the ratior P of the single-phonon crossection to that from para-ortho conversion as given in Eq.~25! as afunction of momentum transfer. The dotted line shows 503r J51

(T50) as given in Eq.~26!. The experimental situation of Ref.corresponds to a momentum transfer between 2 and 4 Å21.

21430

a-r

o

o

ratio of the cross sections at energy gain to those of eneloss does satisfy detailed balance. The ratio of the total csection~counting both energy loss and energy gain! for tran-sitions within the (J51) ground manifold to that due to parto ortho conversion was found at zero temperature to be

r J51~T50!5x

12x

4 j 2~ 12 kr!2

15j 1~ 12 kr!2

. ~26!

Figure 6 shows that this ratio is quite small and therefexperimental observation of this transition~i.e., TE'0.7meV shown in Fig. 4! would be very difficult.

Figure 7 shows the neutron energy-loss spectrum andcalculated total spectrum using the same potential, thecalled WS77 model,15 used by FitzGeraldet al.5 Eventhough the calculated spectrum is wider than the experimtal spectrum, it is still possible to make a one to one corspondence between calculation and experiment as is shby arrows in Fig. 7. The top curve in this figure shows whthe spectrum looks like if the orientational part of the potetial is scaled by about half. The agreement between theand calculations is somewhat better after this arbitrary sing, indicating that the potential used is too anisotropicthe center-of-mass motion of H2 molecule but at the samtime it is too weak for the orientational part of H2 @because itgives too small a result ford, the splitting of the (J51)ground manifold#.

We also tested other potentials commonly used in theerature and these results are shown in Fig. 8. The top curvfrom Novaco’s 6-12 potential which was developed to stuhydrogen on graphite.4 Clearly this potential gives too lowphonon energies and too little splitting of the (J51) levelsfor H2 in solid C60. The other two curves in Fig. 8 are 6-expotentials tabulated in Ref. 15. The spectrum from thesetentials does not agree with experiment either. We asearched the potential parametersA andB for 6-12 andA, B,

shatg-ng

FIG. 7. Neutron energy-loss spectrum~middle! at 4.2 K. Thebottom curve is the result from our calculation using the WSpotential. The dashed and gray lines are the contributions fromtational and vibrational excitations, respectively. The top curvethe spectrum after arbitrarily scalingAl

m by half, indicating that theorientational potential used in our calculations is too anisotropicfar as the center-of-mass motion is concerned.

1-11

otot

tiomn

Vin

outh

thesraheurtiousiooned

ob

icei

noo

ledof

h toenithred

edntly

ec-eWeofen-S77islly

inulele

m-

a-

ol-

ouw

,eth

m of

V.4.8


and C for 6-exp types of potentials. However, we were nable to improve the fit to experiment using these atom-apotentials. Hence it seems that simple atom-atom potendoes not describe the details of the H2-C60 interaction well. Itis an open and important question to find a better potencan reproduce the experimental spectrum better. It is alsimportant to see how well the potentials obtained frodensity-functional theory within local-density approximatiowill do.

Finally, in addition to features observed near 14 meFitzgeraldet al.5 also observed a feature at about 28 meVthe energy gain time-of-flight spectrum. This energy is abtwice that of either the para-ortho conversion energy orenergy of the translational phonon for an H2 molecule in anoctahedral interstitial site. Clearly this feature representsenergy of two excitations, but it was not clear whether thwould be two phonons, one phonon, and one ortho-para tsition, or two ortho-para transitions. In Fig. 9 we show ttemperature dependence of the total intensity of this featThis temperature dependence follows a thermal activawith an energy of about 14 meV. Thus the initial state mconsist of one thermally excited phonon and the transitobserved destroys one thermal phonon and convertsortho molecule~which occurs with temperature-independeprobability x) to a para molecule, thus giving the observenergy of about 28 meV.

VI. CONCLUSIONS

Our results indicate that the coupled phonon-roton prlem is a rich one. For the light molecules of H2 where thesplitting of rotational levels is large compared to most lattvibrational modes, one is in a so-called weak-coupling limwhere the interactions between rotons and translatiophonons can be treated perturbatively. Even at this levelrecovers an interesting structure. Needless to say, we h

FIG. 8. Neutron energy-loss spectrums obtained from varicommonly used intermolecular potentials. For each potentialgive the value ofd, the splitting of the (J51) zero-phonon levelswhich may be compared to the experimentally determined valud50.75 meV~Ref. 5!. Note that the average phonon energies offirst two potentials~top and middle curves! are way off from theobserved value of'14 meV.

21430

tmial

alof

,

te

een-

e.ntnnet

-

talnepe

that the calculations in this paper will motivate more detaiexperiments at higher resolution to elucidate the structurethe roton-phonon spectrum.

We may summarize our main conclusions as follows:• We have presented a systematic perturbative approac

the calculation of the roton-phonon spectrum of hydrogmolecules in confined geometry. Our calculations agree wthe group theoretical analysis for the geometries considehere.

• In a general way, the techniques of this paper~use of theatom-atom potential combined with perturbation theory! mayprove useful to treat hydrogen molecules in other confingeometries, in particular in or on nanotubes. We are curreanalyzing this situation.

• We give a calculation of the expected energy-loss sptrum from hydrogen in C60 in the energy range wherphonons and para-ortho conversion both are important.find that none of the traditional 6-12 and 6-exp typespotentials give good results for the detailed energy depdence of the observed phonon spectrum, although the Wpotential15 we used was definitely the most satisfactory. Ita theoretical challenge to determine a potential which fureproduces the observed spectrum.

• We identify the feature at 28 meV in the energy gaspectrum as consisting of conversion of one ortho molecto a para molecule combined with annihilation of a singphonon. This identification is uniquely indicated by the teperature dependence of this feature.

ACKNOWLEDGMENTS

We acknowledge partial support from the Israel-US Bintional Science Foundation.

APPENDIX A: ATOM-ATOM POTENTIAL AND ALM

We model the interactions between the hydrogen mecule and the surrounding cage of C60 molecules using an

se

e

FIG. 9. Temperature-dependent neutron energy gain spectruH2 in C60 ~the data is taken from Ref. 5!. The inset shows ln(I/I0)versus 1/kT, whereI is the intensity of the feature at about 28 meThe slope of the line indicates an activation energy barrier of 1meV.

1-12

-p

nd

caateror

aco.d

H

hensf

e

heof

al

ofebe

q.

on


atom-atom potential.15 Unless otherwise indicated, all the results reported in this paper are obtained from the sametential 2A/r 61B exp(2Cr) that is used in Ref. 5~whereA55.94 eV Å6, B5678.2 eV, andC53.67 Å21).

We consider two cases depending on the orientatiostate of C60 molecules. When the molecules in the surrouning cage are orientationally disordered, we distribute thebon centers uniformly over the surface of a sphere. This ccorresponds to the octahedral symmetry discussed in theThe resulting integration of the atom-atom potential ovespherical surface is done analytically in Ref. 5 and therefis not given here. For thePa3 symmetry, the C60 moleculesare oriented according to theirPa3 settings and then thetotal potential andAl

m’s are calculated on a mesh points ofcube centered at the octahedral site. Below we derive avenient way to obtain theAl

m’s from the atom-atom potentialWe write the potentialVH-C between a single H atom an

a single C atom as

VH-C5F~r !, ~A1!

wherer is the displacement of the C atom relative to theatom. Thus the interactionV(H2) of a H2 molecule with a Catom can be written as

V~H2!5 (s561

F@~r 21 14 r21srr•n!1/2#, ~A2!

where nowr is the displacement of the C atom relative to tcenter of the H2 molecule whose atoms are at positio6 1

2 rn, wheren is a unit vector specifying the orientation othe molecular axis of the H2 molecule. Then

A2lm5(

s(

iE Y2l

m~ n!* F@~r i21 1

4 r21srr i•n!1/2#dV,

~A3!

wheredV indicates an integration over all orientations ofnand the sum overi is over all neighboring carbon atoms. Wuse this to get

A00[V052Ap(

iE

21

1

F@~r i21 1

4 r21rr ix!1/2#dx.

~A4!

To get theAlm’s for l .0 we write

(s

F@~r i21 1

4 r21srr i•n!1/2#5(L

B2L~r i !Y2L0 ~u r ,n!,

~A5!

where u r ,n is the angle between the vectorsr i and n. Wehave that

B2L~r i !52p(s

E0

p

F@~r i21 1

4 r21srr icosu!1/2#

3Y2L0 ~u!sinudu. ~A6!

Now substitute Eq.~A5! into Eq. ~A3! to get

21430

o-

al-r-sext.ae

n-

A2lm5(

iE Y2l

m~ n!* (L

B2L~r i !Y2L0 ~u r ,n!dV. ~A7!

Using the addition theorem for spherical harmonics20 wehave

A2lm5(

iE Y2l

m~ n!* (L

B2L~r i !A4p/~2L11!

3(n

Y2Ln ~ n!Y2L

n ~ r i !* dV

5(i

B2l~r i !A4p/~2l 11!Y2lm~ r i !* . ~A8!

For A2m we get

A2m52p(

iY2

m~ r i !* E21

1

~3x221!

3F~@r i21 1

4 r21r irx#1/2!dx. ~A9!

For power-law functionsF ~i.e., F;1/r 2n), this integral canbe done analytically.

APPENDIX B: SPHERICAL CAVITY

In this appendix we apply our formalism discussed in ttext to a simple toy model to utilize the main physicsquantum roton-phonon dynamics of a H2 molecule confinedin a spherically symmetric cavity for which the potentidiffers perturbatively from harmonic.

1. Orientationally dependent potential

For a diatomic molecule for which the spherical partthe potential is that of an isotropic spherical oscillator, wtake the orientationally dependent part of the potential to

U~r ,V!

5 f ~r !@~ x sinu cosf1 y sinu sinf1 z cosu!22 13 #,

~B1!

wherex[x/r , y[y/r , and z5z/r .This potential can be written in the canonical form of E

~2! with

A2m5

8p

15f ~r !Y2

m~ r !* . ~B2!

2. Energy shift of the JÄ1 manifold

To evaluate Eq.~16! for the shift in the center of gravityof the J51 zero phonon levels, we need the wave functiof the ground state, namely

c0~r !5ae2r 2/(4s2). ~B3!

wherea5s23/2(2p)23/4. Also we can write the two-phononexcited states~for spherical symmetry! as

1-13

W

sius

otn

he

r-

ni-

ni-


c2,m(L52)5b~r /s!2Y2

m~ r !e2r 2/(4s2), ~B4!

for m522,21,0,1,2, and for angular momentumL52 andwhereb5s23/2A2/15(2p)21/4. The sixth two-phonon stateis ans-wave state, whose wave function we do not need.will also need theL52 (d-wave! four-phonon states whichare

c4,m(L52)5g@~r /s!427~r /s!2#Y2

m~ r !e2r 2/(4s2), ~B5!

for m522,21,0,1,2, and whereg5s23/2A1/105(2p)21/4.Now we assume the following polynomial fit forf (r ):

f ~r !5g2~r /s!21g4~r /s!41g6~r /s!6. ~B6!

We then find

u^0uA2m~r !uc2,m

(L52)&u54Ap

15ug217g4163g6u[4Ap

15G

~B7!

and

u^0uA2m~r !uc4,m

(L52)&u54A14p

15ug4118g6u. ~B8!

Thus we have

DE524

15

~g217g4163g6!2

\v2

28

15

~g4118g6!2

\v.

~B9!

From numerics we get

A20~z!50.07226r 410.0348r 6. ~B10!

For convenience we takes50.25 Å. ThenA20 is of the form

of Eqs.~B2! and ~B6! with

g450.07226s4, g650.0348s6, ~B11!

g450.07226/25650.282 meV, ~B12!

g650.0348/409650.0084 meV. ~B13!

We evaluate the first and second terms on the right-handof Eq. ~B9! to be 0.115 and 0.025 meV, respectively. Thwe have

DE50.14 meV. ~B14!

Notice thatDE is a very strong function ofs. For instance,if you take s50.26 Å, you get g450.33 meV andg650.011 meV in which caseDE50.19 meV.

3. Effect of translation-rotation coupling on the JÄ1 one-phonon manifold

In the absence of coupling between translations and rtions we characterize the single-phonon states by phoangular momentumLP , so that

21430

e

de

a-on

uLP561&57S x6 iy

s Dc0~r !, uLP50&5~z/s!c0~r !.

~B15!

We now wish to include the effect of the perturbation of tform of Eq.~2! when the coefficients are as in Eq.~B2!, withf (r ) given by Eq.~B6!. We know that states are now chaacterized by the total angular momentumK5LP1J. So theenergy of theK52 manifold is given by

E~K52!5^LP51,Jz51uU~r ,V!uLP51,Jz51&

521

15

E dr ux2 iy

su2e2r 2/(2s2) f ~r !

3z22r 2

r 2

E dr ux2 iy

su2e2r 2/(2s2)

3^Jz51u~3Jz222!uJz51&

5

2E dr r 2f ~r !e2r 2/(2s2)

225s2E dre2r 2/(2s2)

52

15G, ~B16!

wheref (r ) andG are defined in Eqs.~B6! and~B7!, respec-tively.

Similarly, one can evaluate the Hamiltonian in the mafold of statesf15uLP51,Jz50& and f25uLP50,Jz51&.The Hamiltonian matrix in this basis is found to be

H5GF 24

15

2

5

2

52

4

15G . ~B17!

This matrix has an eigenvalue215 G which is associated withthe K52, Kz51 state and the new eigenvalue for theK51 manifold,E(K51)52 2

3 G.Similarly, one can evaluate the Hamiltonian in the ma

fold of states f15uLP51,Jz521&, f25uLP50,Jz50&,and f35uLP521,Jz51&. The Hamiltonian matrix in thisbasis is found to be

H5G32

152

2

5

4

5

22

5

8

152

2

5

4

52

2

5

2

154 . ~B18!

In this manifold we reproduce the eigenvalues forK52 andK51. The new eigenvalue isE(K50)5 4

3 G.

1-14

nt

ng

th

l-cos.rethrs

ero

eu-

-

e


APPENDIX C: NEUTRON-SCATTERING CROSS SECTION

1. General formulation

Following Elliott and Hartmann,21 we write

V~r2rn!52p\

m0d~r2rn!@b1b8~s•I !#, ~C1!

wherem0 is the neutron mass,s is the neutron spin,r is thecoordinate of the proton,rn the coordinate of the neutron,I isthe proton spin, andb andb8 are the coherent and incoherescattering lengths for this scattering. Sinceb is very small,we drop that term from now on. The differential scattericross section is

]2s

]V]E5

k8

k (i f

Pid~E2Ei1Ef !uVu2, ~C2!

whereE5\2@(k8)22k2#/(2m0), Pi is the Boltzmann prob-ability, and the sum is over all states of the system. Herepotential is

V5b8(j a

ei k•Rj ,as•I j ,a , ~C3!

wherek5k82k, j labels molecules anda51,2 the atomswithin a molecule. Performing the sum overa we get

V5b8(j

ei k•Rj$~s•I j !cos~ 12 k•r!

1 i sin~ 12 k•r!@s•~ I j 12I j 2!#%, ~C4!

where I j5I j 11I j 2. The first term acts only on ortho moecules because for paras the total spin is zero and the seterm causes transitions between ortho and para moleculewe write the scattering cross section as the sum of thterms, the first of which represents scattering from an ormolecule and others ortho-para conversion or the reveThus

]2s

]V]E5

k8

k@NxS1→11NxS1→01N~12x!S0→1#,

~C5!

whereN is the total number of molecules andx is the orthoconcentration. Because there are no correlations betwnuclear spins each cross section is actually a sum over csections for each molecule:

Sb5(j

Sb j , ~C6!

wherej labels the molecule,b is 0→1, etc., and

S0→1,j5 (Ji50,Jf51

Pid~E2Ei1Ef !u^ f ub8ei k•Rjsin~ 12 k•r!

3@s•~ I j 12I j 2!#u i &T2,

21430

e

ndSoeoe.

enss

S1→0,j5 (Ji51,Jf50

Pid~E2Ei1Ef !u^ f ub8ei k•RjsinS 1

2k•rD

3@s•~ I j 12I j 2!#u i &T2,

S1→1,j5 (Ji51,Jf51

Pid~E2Ei1Ef !u^ f ub8ei k•RjcosS 1

2k•rD

3@s•I #u i &T2, ~C7!

where the sums are over states for the fixed species~ortho orpara! of molecule as indicated and the subscriptT indicatesthat the wave functions include nuclear-spin functions.

Now we perform the sum over the spin states of the ntron and proton to obtain the results

S0→1,j534 ~b8!2 (

Ji50,Jf51Pid~E2Ei1Ef !

3u^ f uei k•Rjsin~ 12 k•r!u i &2,

S1→0,j514 ~b8!2 (


3u^ f uei k•Rjsin~ 12 k•r!u i &2,

S1→1,j512 ~b8!2 (


3u^ f uei k•Rjcos~ 12 k•r!u i &2, ~C8!

where now statesu f & andu i & no longer include nuclear spinwave functions. We write

ei k•Rj5ei k•(Rj(0)

1uj )'ei k•Rj(0)

e2(1/2)(k•uj )2@11 i ~k•uj !#

'e2Wei k•Rj(0)

@11 i ~k•uj !#, ~C9!

whereRj(0) is the equilibrium value ofRj andW' 1

6 k2û2&[ 1

6 k2ûx21uy

21uz2&. Since spherical harmonics of degre

higher than two do not affect the manifolds ofJ50 or J51, we now use

cos~ 12 k•r!5 j 0~ 1

2 kr!24p j 2~ 12 kr!(

nY2

n~ k!* Y2n~ r!

~C10!

and

sin~ 12 k•r!54p j 1~ 1

2 kr!(n

Y1n~ k!* Y1

n~ r!. ~C11!

We expand in displacements to get

1-15

ntth

q

h

r

ec-

tonde-

tem

atent.

s to


S0→1,j5~4p!2A (Ji50,Jf51

Pid~E2Ei1Ef !^ f u@11 i k•uj #

3(n

Y1n~ k!* Y1

n~ r!u i &^ i u@12 i k•uj #

3(m

Y1m~ k!Y1

m~ r!* u f &, ~C12!

S1→0,j5~4p!2B (Ji51,Jf50

Pid~E2Ei1Ef !^ f u@11 i k•uj #

3(n

Y1n~ k!* Y1

n~ r!u i &^ i u@12 i k•uj #

3(m

Y1m~ k!Y1

m~ r!* u f &, ~C13!

where A5 34 e22W@b8 j 1( 1

2 kr)#2, B5 14 e22W@b8 j 1( 1

2 kr)#2,and

S1→1,j512 ~b8!2 (

Ji51,Jf51Pid~E2Ei1Ef !^ f u@11 i k•uj #

3F j 0~ 12 kr!24p j 2~ 1

2 kr!(n

Y2n~ k!* Y2

n~ r!G u i &3^ i u@12 i k•uj #F j 0~ 1

2 kr!

24p j 2~ 12 kr!(

mY2

m~ k!Y2m~ r!* G u f &. ~C14!

Since the phonon energy is much larger than the orietional energy, we may classify contributions accordingnumber of phonons that are involved. In the notation of E~C6! we write

Sb5S b(0)1S b

(1) , ~C15!

whereS b(0) corresponds to a zero-phonon process andS b

(1) toa process in which one phonon is created or destroyed. T

S 0→1(0) 5~4p!2A (

Ji50,Jf51Pid~E2Ei1Ef !(

mn^ f uY1

n~ r!u i &

3^ i uY1m~ r!* u f &Y1

n~ k!* Y1m~ k!, ~C16!

S 0→1(1) 5~4p!2A (


3 (mnab

^ f uua* Y1n~ r!u i &^ i uub* Y1

m~ r!* u f &

3kakbY1n~ k!* Y1

m~ k!. ~C17!

Here and below we use spherical components of a vectov:v6157(vx6 ivy)/A2 andv05vz . Similarly

21430

a-e.

us

S 1→0(0) 5~4p!2B (


mn^ f uY1

n~ r!u i &

3^ i uY1m~ r!* u f &Y1

n~ k!* Y1m~ k!, ~C18!

S 1→0(1) 5~4p!2B (


3(mn

^ f uua* Y1n~ r!u i &^ i uub* Y1

m~ r!* u f &

3kakbY1n~ k!* Y1

m~ k!, ~C19!

S1→1(0) 5~4p!2C (


m,n^ f uY2

m~ r!u i &

3^ i uY2n~ r!u f &Y2

m~ k!* Y2n~ k!* , ~C20!

whereC5 12 e22W@b8 j 2( 1

2 kr)#2 and

S 1→1(1) 5D0 (


ab^ f uua* u i &

3^ i uubu f &ka* kb14pD1 (Ji51,Jf51

Pid~E2Ei1Ef !

3 (mab

^ f uua* Y2m~ r!u i &^ i uub* u f &kakbY2

m~ k!*

14pD1 (Ji51,Jf51

Pid~E2Ei1Ef ! (mab

^ f uua* u i &

3^ i uub* Y2m~ r!u f &kakbY2

m~ k!*

1~4p!2D2 (Ji51,Jf51

Pid~E2Ei1Ef !

3 (mnab

^ f uua* Y2n~ r!* u i &

3^ i uub* Y2m~ r!u f &kakbY2

m~ k!* Y2n~ k!

5 (n51

4

S1→1;n(1) , ~C21!

where Dn5 12 (21)n(b8)2e22Wj 0( 1

2 kr)22nj 2( 12 kr)n and

S1→1;n(1) is the contribution to the one-phonon ortho cross s

tion from thenth term in the first equality.Note the existence of terms in which a phonon and a ro

are created, the system evolves, and finally a phonon isstroyed. This type of process can only occur when the syssupports roton-phonon interactions. All the termsS1→1;n

(1) cor-respond approximately to the phonon energy.

2. Powder average at low temperature

Here we restrict attention to the energy-loss spectrumlow temperature when there are no thermal phonons presAlso, we now take the powder average. This correspond

1-16

entitiin

th

e,

fo

lo

ec-he

we

ednte-

orn isethe

sh


actual experimental situation in Ref. 5, but would also breasonable approximation to take account of the differeoriented symmetry axes of the various octahedral interstsites. Below we calculate the cross sections for the followprocesses;~a! energy loss by conversion,~b! energy gain byconversion,~c! single-phonon energy loss, and finally~d!zero-phonon transition from (J51,M50) to (J51,M561).

a. Energy loss by conversion

We have

^S 0→1(0) &5~4p!A (


3(m

^ f uY1m~ r!u i &^ i uY1

m~ r!* u f &, ~C22!

where^ & indicates a powder average. The initial state isJ50,Jz50 zero-phonon state, which we write as

c i5(r

ci~r !ur ;J50;Jz50&, ~C23!

where ci(r ) is the amplitude of the wave function at thmesh pointr . The final state is aJ51 zero-phonon statewhich we similarly write as

c f ,m5(r ,M

cf ,m~r ,M !ur ;J51;Jz5M & ~C24!

and whose energy relative to theJ50 state is

Ef ,m5Ec1Em . ~C25!

If EL52E is the energy loss, we may write

^S 0→1(0) ~EL!&5A(

md@EL2~Ec1Em!#

3(m

U(r

cf ,m~r ,m!* ci~r !U2

. ~C26!

To a good approximation the zero-phonon wave functionsJ51 can be chosen to be composed of a single value ofJz .Thus we may label the wave functions so thatm5m:

^S 0→1(0) ~EL!&5A(

md@EL2~Ec1Em!#

3U(r

cf ,m~r ,m!* ci~r !U2

. ~C27!

The corresponding integrated intensity is

I (0)[E dEL^S 0→1(0) &5A(

mU(

rcf ,m~r ,m!* ci~r !U2

.

~C28!

The inner product of theJ50 wave function and the spatiapart of theJ51 zero-phonon states is essentially unity. S

21430

alyalg

e

r

I (0)53A. ~C29!

b. Energy gain by conversion

Here we give a similar analysis of the energy-gain sptrum at low temperature due to ortho-para conversion. Tderivation is similar to that for para-ortho conversion soonly quote the results:

^S 1→0(0) ~EL!&5B(

mPmd@E2~Ec2Em!#

3U(r

ci ,m~r ,m!* cf~r !U2

, ~C30!

wherePm is the probability that theJ51,Jz5m state is oc-cupied and the role of initial and final states is interchangfrom the para to ortho processes. The corresponding igrated intensity is

I (0)[E dE^S 1→0(0) &5B(

mPmU(

rci ,m~r ,m!* cf~r !U2

5B.

~C31!

SinceB53A we see that ratio of the total cross section fortho to para conversion to that of para to ortho conversio(12x) to x, wherex is the ortho concentration. Normally thratio of energy gain to energy-loss cross sections followsBoltzmann factor. Here, the populations are set byx ratherthan by the temperature.

c. Single-phonon energy loss

We have the powder average ofS1→1;1(1) of Eq. ~C21! as

^S1→1;1(1) &5

1

3k2D0 (


Lu^ f uuLu i &u2

~C32!

and the corresponding integrated intensity is

I 1(1)[E dEL^S1→1;1

(1) ~EL!&51

3k2û2&D0 . ~C33!

In terms of the amplitudes of the wave function on the mepoints, the above result is

^S1→1;1(1) &5

1

3k2D0(

i , fPid~EL2Ef1Ei !

3(L

U(M ,r

ci~M ,r !cf~M ,r !r LU2

. ~C34!

Also we obtain the powder average ofS1→1;2(1) andS1→1;3

(1)

of Eq. ~C21! as

1-17

ca

ma

h

ro-

these


^S1→1;2(1) ~EL!&5^S1→1;3

(1) ~EL!&*

522

5A8p

15D1k2(

i , fPid~EL2Ef1Ei !

3 (M ,M8

C~112;M ,M 8!~21!M1M8

3^ f uu2MT2M1M8~J!u i &^ i uu2M8u f &,

~C35!

where theT2M(J) are the operator equivalents of the spheri

harmonics:22

T262~J!5A 15

32pJ6

2 ,

T261~J!57A 15

32p~JzJ61J6Jz!,

T20~J!5A 5

16p~3Jz

222!. ~C36!

Here

^ i uu2M8u f &5(m,r

ci~m,r !* cf~m,r !r 2M8 . ~C37!

and

^ f uu2MT2M1M8~J!u i &5

5

A8p(m,r

ci~m,r !cf~m1M

1M 8,r !* r 2MC~121;m,M1M 8!,

~C38!

where we used

^J51;Jz5M uT2L~r!uJ51;Jz5M 8&

55

A8pdM ,L1M8C~121;M 8,L !. ~C39!

We have the contributions to the integrated intensity

I 2(1)5I 3

(1)* 5E dEL^S1→1;2(1) ~EL!&

522

5A8p

15D1k2(

i fPi (

M ,M8C~112;M ,M 8!

3~21!M1M8^ i uu2M8u f &^ f uu2MT2M1M8~J!u i &.

~C40!

Here the sum over final states should be restrictedsingle-phonon states. Higher energy states make only a scontribution to this sum. So we make the closure approximtion that the sum overu f & extends over all states, in whiccase

21430

l

toall-

I 2(1)52

2

5A8p

15D1k2(

iPi (

M ,M8C~112;M ,M 8!

3~21!M^ i uuM8* u2MT2

M1M8~J!u i &. ~C41!

As illustrated by Eq.~20!, the initial stateu i & is dominantlycomprised of a single value ofJz . Thus in Eq.~C41! M1M 850 dominates. In addition, the system is nearly isotpic. Then (MC(112;M ,2M )(21)Muu2Mu250. So, to agood approximation,

I 2(1)1I 3

(1)50. ~C42!

We have made several approximations, but our result fortotal integrated intensity will not be much affected by theapproximations.

Similarly, we get

^S1→1;4(1) ~EL!&5

16p

75k2D2 (

i , f ,m,MPid~EL2Ef

1Ei !u^ f uuMT2m~J!u i &u2

232p

25A21D2k2 (

i , f ,n,M ,M8Pid~EL2Ef

1Ei !C~112;M ,M 8!C~222;M1M 8,n!

~21!M8^ f uu2MT22n~J!u i &

3^ f uuM8T22M2M82n~J!u i &* . ~C43!

We now evaluate the integrated intensity,

I 4(1)[E dEL^S1→1;4

(1) ~EL!&[t11t2 , ~C44!

where

t1516p

75k2D2 (

i , f ,m,MPi^ i uu2MT2

2m~J!u f &

3^ f uuMT2m~J!u i &~21!M1m. ~C45!

Making the closure approximation this is

t1516p

75k2D2 (

i ,m,MPi^ i uu2MuMT2

2m~J!T2m~J!u i &~21!m1M

52

3k2D2(

iPi^ i uu2MuMu i &~21!M5

2

3k2D2û

2&.

~C46!

Similarly

t25232p

25A21D2k2 (

i ,M ,M8n

PiC~112;M ,M 8!C~222;M

1M 8,n!~21!M1n^ i uuM* T2M1M81n~J!u2MT2

2n~J!u i &.

~C47!

1-18

ha

hew

rec

by

ssto

n

ra


Again, we treatu i & as having a single value ofJz , so thatM1M 850. Also we again assume spatial isotropy, so t(MC(112;M ,2M )(21)Muu2Mu250. Thent250.

d. Zero-phonon ortho cross section

Finally we consider the zero-phonon contribution to tortho to ortho cross section. Taking the powder average,have

^S1→1(0) &54pC (

Ji51(

Jf51Pid~E2Ei1Ef !

3(m

^ f uY2m~ r!u i &^ i uY2

2m~ r!u f &~21!m.

~C48!

If d is the energy of theJz561 levels relative to theJz50 level and if the Boltzmann probability of the stateum& isPm , then we may write

^S1→1(0) &54pCP0d~E1d! (

m561@ u^muY2

1~ r!u0&u2

1u^muY221~ r!u0&u2#14pCd~E2d!

3 (m561

Pm@ u^muY21~ r!u0&u21u^muY2

21~ r!u0&u2#,

~C49!

which gives

^S1→1(0) &5 6

5 CP0d~E1d!1 65 CP1d~E2d!. ~C50!

The ratio of cross sections at energy gain to those of eneloss does satisfy detailed balance because within the sp(J51) we do maintain thermal equilibrium.

e

n,

y,

Mx

21430

t

e

gyies

3. Intensity ratios

We develop an expression for the ratior P , defined to bethe integrated intensity due to phonon creation dividedthat due to para-ortho conversion. Using the results forI (n)

obtained above we have

r P5xEdEL^S1→1

(1) ~EL!&

~12x! EdEL^S0→1(0) ~EL!&

5x@ I 1

(1)1I 4(1)#

~12x!I (0)

5

13 k2û2&D01 2

3 k2û2&D2

3A S x

12xD5

2

27k2û2&

j 0~ 12 kr!212 j 2~ 1

2 kr!2

j 1~ 12 kr!2 S x

12xD .

~C51!

The ratior C, defined to be the ortho to para conversion crosection in energy gain to that in energy loss due to paraortho conversion, is given by

r C5x

12x. ~C52!

Finally, we haver J51, defined to be the total cross sectio~counting both energy loss and energy gain! for transitionswithin theJ51 ground manifold divided by that due to pato ortho conversion, as

r J515S x

12xD 65 C~P01P1!

3A5~P01P1!r J51~T50!,

~C53!

where

r J51~T50!5S x

12xD 4 j 2~ 12 kr!2

15j 1~ 12 kr!2

. ~C54!

.x,

T.J.n,

m,an

.

lved to

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Rotational and vibrational dynamics of interstitial …Rotational and vibrational dynamics of interstitial molecular hydrogen T. Yildirim National Institute of Standards and Technology,